Nullity conditions in paracontact geometry

The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition \eqref{paranullity} below, for some real numbers $\tilde\kappa$ and $\tilde\mu$). This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was recently defined in \cite{MOTE}. In this paper we show in fact that there is a kind of duality between those manifolds and contact metric $(\kappa,\mu)$-spaces. In particular, we prove that, under some natural assumption, any such paracontact metric manifold admits a compatible contact metric $(\kappa,\mu)$-structure (eventually Sasakian). Moreover, we prove that the nullity condition is invariant under $\mathcal{D}$-homothetic deformations and determines the whole curvature tensor field completely. Finally non-trivial examples in any dimension are presented and the many differences with the contact metric case, due to the non-positive definiteness of the metric, are discussed.Comment: Different. Geom. Appl. (to appear

Contact metric (κ,μ)(\kappa,\mu)-spaces as bi-Legendrian manifolds

We regard a contact metric manifold whose Reeb vector field belongs to the $(\kappa,\mu)$-nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric $(\kappa,\mu)$-spaces in terms of a canonical connection which can be naturally defined on them.Comment: To appear on Bull. Austral. Math. So

Sasaki-Einstein and paraSasaki-Einstein metrics from (\kappa,\mu)-structures

We prove that any non-Sasakian contact metric (\kappa,\mu)-space admits a canonical \eta-Einstein Sasakian or \eta-Einstein paraSasakian metric. An explicit expression for the curvature tensor fields of those metrics is given and we find the values of \kappa and \mu for which such metrics are Sasaki-Einstein and paraSasaki-Einstein. Conversely, we prove that, under some natural assumptions, a K-contact or K-paracontact manifold foliated by two mutually orthogonal, totally geodesic Legendre foliations admits a contact metric (\kappa,\mu)-structure. Furthermore, we apply the above results to the geometry of tangent sphere bundles and we discuss some topological and geometrical properties of (\kappa,\mu)-spaces related to the existence of Eistein-Weyl and Lorentzian Sasakian Einstein structures

Bi-paracontact structures and Legendre foliations

We study almost bi-paracontact structures on contact manifolds. We prove that if an almost bi-paracontact structure is defined on a contact manifold $(M,\eta)$, then under some natural assumptions of integrability, $M$ carries two transverse bi-Legendrian structures. Conversely, if two transverse bi-Legendrian structures are defined on a contact manifold, then $M$ admits an almost bi-paracontact structure. We define a canonical connection on an almost bi-paracontact manifold and we study its curvature properties, which resemble those of the Obata connection of an anti-hypercomplex (or complex-product) manifold. Further, we prove that any contact metric manifold whose Reeb vector field belongs to the $(\kappa,\mu)$-nullity distribution canonically carries an almost bi-paracontact structure and we apply the previous results to the theory of contact metric $(\kappa,\mu)$-spaces.Comment: To appear on: Kodai Mathematical Journa

A preliminary mechanical property and stress corrosion evaluation of VIM-VAR work strengthened and direct aged Inconel 718 bar material

This report presents a preliminary mechanical property and stress corrosion evaluation of double melted (vacuum induction melted (VIM), and vacuum arc remelted (VAR)), solution treated, work strengthened and direct aged Inconel 718 alloy bar (5.50 in. (13.97 cm) diameter). Two sets of tensile specimens, one direct single aged and the other direct double aged, were tested at ambient temperature in both the longitudinal and transverse directions. Longitudinal tensile and yield strengths in excess of 200 ksi (1378.96 MPa) and 168 ksi (1158.33 MPa), respectively, were realized at ambient temperature, for the direct double aged specimen. No failures occurred in the single or double edged longitudinal and transverse tensile specimens stressed to 75 and 100 percent of their respective yield strengths and exposed to a salt fog environment for 180 days. Tensile tests performed after the stress corrosion test showed no mechanical property degradation

A mechanical property and stress corrosion evaluation of MP 35N multiphase alloy

Mechanical properties and stress corrosion evaluation of MP 35N multiphase alloy at cryogenic temperatur

The Stress Corrosion Resistance and the Cryogenic Temperature Mechanical Behavior of 18-3 Mn (Nitronic 33) Stainless Steel Parent and Welded Material

The ambient and cryogenic temperature mechanical properties and the ambient temperature stress corrosion results of 18-3 Mn (Nitronic 33)stainless steel, longitudinal and transverse, as received and as welded (TIG) material specimens manufactured from 0.063 inch thick sheet material, were described. The tensile test results indicate an increase in ultimate tensile and yield strengths with decreasing temperature. The elongation remained fairly constant to -200 F, but below that temperature the elongation decreased to less than 6.0% at liquid hydrogen temperature. The notched tensile strength (NTS) for the parent metal increased with decreasing temperature to liquid nitrogen temperature. Below -320 F the NTS decreased rapidly. The notched/unnotched (N/U) tensile ratio of the parent material specimens remained above 0.9 from ambient to -200 F, and decreased to approximately 0.65 and 0.62, respectively, for the longitudinal and transverse directions at liquid hydrogen temperature. After 180 days of testing, only those specimens exposed to the salt spray indicated pitting and some degradation of mechanical properties

Relating Scattering Amplitudes in Bosonic and Topological String Theories

A formal relationship between scattering amplitudes in critical bosonic string theory and correlation functions of operators in topological string theory is found.Comment: 9 page